Multiplicity of Positive and Nodal Solutions for Nonhomogeneous Elliptic Problems in Unbounded Cylinder Domains
© Tsing-San Hsu 2009
Received: 13 March 2009
Accepted: 7 May 2009
Published: 9 June 2009
For the homogeneous case, that is, , Zhu  has established the existence of a positive solution and a nodal solution of problem (1.2) in provided satisfies in and as for some positive constants and . More recently, Hsu  extended the results of Zhu  with to an unbounded cylinder . Let us recall that, by a nodal solution we mean the solution of problem (1.2) with change of sign.
For the nonhomogeneous case ( ),Adachi and Tanaka  have showed that problem (1.2) has at least four positive solutions in for and satisfy some suitable conditions, but we place particular emphasis on the existence of nodal solutions. More recently, Chen  considered the multiplicity results of both positive and nodal solutions of problem (1.2) in . She has showed that problem (1.2) has at least two positive solutions and one nodal solution in when and satisfy some suitable assumptions.
In the present paper, motivated by  we extend and improve the paper by Chen . We will deal with unbounded cylinder domains instead of the entire space and also obtain the same results as in . Our arguments are similar to those in [5, 6], which are based on Ekeland's variational principle .
Now, we state our main results.
where , , and is a bounded domain in . They found that (1.10) possesses infinitely many solutions. More recently, Tarantello  proved that if is the critical Sobolev exponent and satisfying suitable conditions, then (1.10) admits two solutions. For the case when is an unbounded domain, Cao and Zhou , Cîrstea and Rădulescu , and Ghergu and Rădulescu  have been investigated the analogue equation (1.10) involving a subcritical exponent in . Furthermore, Rădulescu and Smets  proved existence results for nonautonomous perturbations of critical singular elliptic boundary value problems on infinite cones.
This paper is organized as follows. In Section 2, we give some notations and preliminary results. In Section 3, we will prove Theorem 1.1. In Section 4, we establish the existence of nodal solutions.
In this paper, we always assume that is an unbounded cylinder domain or . Let for , and let be the first positive eigenfunction of the Dirichlet problem in with eigenvalue unless otherwise specified. We denote by and ( ) universal constants, maybe the constants here should be allowed to depend on and , unless some statement is given. Now we begin our discussion by giving some definitions and some known results.
here and from now on, we omit " " and " " in all the integration if there is no other indication. It is well known that is of in and the solutions of problem (1.2) are the critical points of the energy functional (see Rabinowitz ).
Let us introduce the problem at infinity associated with problem (1.2) as
Furthermore, from Hsu  we can deduce that for any there exist positive constants such that, for all ,
We end this preliminaries by the following definition.
3. Proof of Theorem 1.1
In this section, we will establish the existence of two positive solutions of problem (1.2).
Apply Lemmas 3.1, 3.2, 3.3, and Ekeland variational principle , and we can establish the existence of the first positive solution.
Modifying the proof of Chen [4, Proposition 2.5]. Here we omit it.
To give a proof of Proposition 3.6, we need to establish some lemmas.
Now, we give the proof of Proposition 3.6.
The Proof of Proposition 3.6
Thus from (3.26) and (3.32)–(3.35), we obtain (3.13). This completes the proof of Proposition 3.6.
Hence . So we can always take . By the maximum principle for weak solutions (see Gilbarg and Trudinger ) we can show that if , then in .
The proof of Theorem 1.1
By Propositions 3.4 and 3.10, we obtain the conclusion of Theorem 1.1.
4. Existence of Nodal Solution
Then we have
The proof is almost the same as that in Tarantello [6, Proposition 3.1] .
To prove (4.7), we only need to estimate for and . First, it is obvious that the structure of guarantees the existence of (independent of large) such that , for all . On the other hand, for , since is continuous in , there exists small enough such that
This completes the proof of Lemma 4.2.
It is obvious that is closed. Exactly as in the proof of [6, Proposition 3.2], by means of Ekeland's principle, we derive a -sequence for . In particular, we have , for some constants and . Thus, we can take a subsequence, also denoted by , such that weakly in . We start by showing that .
We claim that . Indeed, we assume is bounded below, as above, (4.28) and (4.32) imply , contradicting (4.31). In the same way, if , we can also prove . Hence we have or ; that is, or . By assumptions and , we conclude that .
The Proof of Theorems 1.2–1.4
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